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Home » Thus, the expanded form is $ oxed8a^3 - 27b^3 $. - Blask

Thus, the expanded form is $ oxed8a^3 - 27b^3 $. - Blask

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Since 17 divides \( R(3) \) and does not divide \( R(1), R(2), R(4), \ldots \), and the gcd must divide all values, but 17 does not divide all, we instead consider the gcd of the entire set.

📌 Compute \( \gcd(6, 17) = 1 \), since 6 and 17 are coprime. Any common divisor of all terms must divide 6 and 17. But \( \gcd(6, 17) = 1 \), so the overall gcd is 1.

📌 Therefore, the greatest common divisor of all \( R(t) \) for \( t = 1 \) to \( 10 \) is:

📌 An epidemiologist tracking a disease outbreak observes that the total number of cases over two consecutive weeks is 144, and the greatest common divisor of the weekly case counts is as large as possible. What is the largest possible value of \( \gcd(a, b) \) if \( a + b = 144 \) and \( a, b \) are positive integers?

📌 Let \( a + b = 144 \), and let \( d = \gcd(a, b) \). Then we can write \( a = d \cdot m \), \( b = d \cdot n \), where \( m \) and \( n \) are coprime positive integers. Then:

📌 a + b = d(m + n) = 144.

📌 Thus, \( d \) must be a divisor of 144, and \( m + n = \frac{144}{d} \). Since \( m \) and \( n \) are coprime positive integers, \( m + n \geq 2 \), so \( \frac{144}{d} \geq 2 \Rightarrow d \leq 72 \).

📌 To maximize \( d \), we take the largest divisor of 144 such that \( \frac{144}{d} \geq 2 \) and there exist coprime \( m, n \) with \( m + n = \frac{144}{d} \).

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\(x = \frac{50 \pm 43.13}{8}\).
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#### 0.86
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New height \(= 10 + 0.5 \times 10 = 15\) cm.
New radius \(= 3 - 0.2 \times 3 = 2.4\) cm.
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